Question: Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$.  Express your answer as a single fraction.
Explanation: First, we find a common denominator, which in this case is the least common multiple of $2$ and $3$, or $6$.  Thus we have $$\frac{2(x+1)}{6}+\frac{3(2-3x)}{6}=\frac{2(x+1)+3(2-3x)}{6}.$$Simplifying the numerator, we get $$\frac{2x+2+6-9x}{6}= \boxed{\frac{8-7x}{6}}.$$